Periodic laminar structures with a period $(\mathrm{\Lambda })$ much smaller than the wavelength $(\lambda )$ of propagating light behave, with respect to the average electromagnetic field, as homogeneous metamaterials [1]. This homogenized metamaterial is frequently characterized as an isotropic medium with a different equivalent refractive index for each polarization, or more generally, as a form birefringent anisotropic medium [2]. These approaches have long been studied in free-space optics [3,4], enabling, for example, the design of half-wave and quarter-wave plates [5,6].

More recently, metamaterial nanostructures have been incorporated into silicon-on-insulator (SOI) waveguides, in the form of subwavelength gratings (SWGs) [7–9]. In SWG devices, a controllable refractive index is synthesized by adjusting the relative amount of silicon per period, i.e., the duty cycle (DC). By approximating the SWG as an isotropic homogeneous material, a wide range of high-performance integrated devices has been demonstrated, including fiber-to-chip grating couplers [10–12], edge couplers [7,13,14], waveguide couplers [15,16], Bragg filters [17,18], multimode bends based on transformation optics [19], and evanescent field waveguide sensors [20,21]. Simple anisotropic models have enabled further applications of SWG waveguides, such as ultra-broadband devices [22,23], polarization management [24,25], or evanescent field confinement [26,27]. However, by changing the duty cycle, both polarizations are affected and can also hamper device fabrication due to aspect ratio dependent etching and proximity effects [28].

In this work, we propose, analyze, and experimentally demonstrate a technique to control the anisotropy of an SWG structure. We show that tilting the silicon segments, as shown in Fig. 1, mainly affects the in-plane (TE) modes, with little impact on the out-of-plane (TM) modes, while maintaining a constant duty cycle and minimum feature size. Therefore, the practical advantages of tilted SWG structures are apparent for applications in on-chip polarization management, where controlling waveguide birefringence is of fundamental importance. We envision that the proposed technique will enable development of nonbirefringent waveguides and novel polarization management devices (e.g., polarization beam splitters based on phase matching).

 

Fig. 1. Scanning electron microscope (SEM) image of a tilted SWG waveguide. The waveguide parameters (width, period, duty cycle, and tilt angle) are $w_c=1\mathrm{\mu m}$, $\mathrm{\Lambda }=0.25\mathrm{\mu m}$, $\mathrm{DC}=0.5$, $\theta =45°$. The device was fabricated in 220 nm thick Si layer on a 3 μm thick layer of buried oxide (BOX). A 3 μm thick upper ${\mathrm{SiO}}_2$ cladding was deposited by plasma-enhanced chemical vapor deposition after the SEM image was taken.

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To gain insight into the physics of tilted SWG structures, let us first revisit the properties of a laminar periodic structure. Specifically, consider a periodic structure with period $\mathrm{\Lambda }$, composed of two transversally infinite materials with refractive indexes $n_1$ and $n_2$, and thicknesses $a=\mathrm{DC}·\mathrm{\Lambda }$ and $b=(1-\mathrm{DC})·\mathrm{\Lambda }$, respectively [see Fig. 2(a)]. For the purpose of illustration, typical values for SWG waveguides in the SOI platform are considered, i.e., $\mathrm{\Lambda }=0.2\mathrm{\mu m}$, $\mathrm{DC}=0.5$, $n_1=3.476$, $n_2=1.444$, and ${\lambda }_0=1.55\mathrm{\mu m}$. A plane wave propagating along this structure with an angle $\varphi$ with respect to the $z$ axis is described by its $k$ vector $\overrightarrow{k}(\varphi )=k_x\widehat{x}+k_z\widehat{z}$, where $\varphi =\mathrm{atan}(k_x/k_z)$ and $k_x$, $k_z$ are related via the dispersion equation [29]

 

Fig. 2. (a) Laminar periodic structure comprising two materials with refractive indexes $n_1$ and $n_2$ and thicknesses $a=\mathrm{DC}·\mathrm{\Lambda }$ and $b=(1-\mathrm{DC})·\mathrm{\Lambda }$. (b) Anisotropic material described by the tensor $\mathit{ϵ}=\mathrm{diag}[n_{xx}^2,n_{yy}^2,n_{zz}^2]$, which characterizes the propagation properties of a laminar periodic structure. (c) Comparison of the $k$ vector of a laminar structure and an anisotropic homogeneous metamaterial for $n_1=3.476$, $n_2=1.444$, $\mathrm{\Lambda }=0.2\mathrm{\mu m}$, $\mathrm{DC}=0.5$, and ${\lambda }_0=1.55\mathrm{\mu m}$.

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This behavior is closely related to that of a homogeneous but anisotropic uniaxial medium described by the dielectric permittivity tensor $\mathit{ϵ}=\mathrm{diag}[n_{xx}^2,n_{yy}^2,n_{zz}^2]$, illustrated in Fig. 2(b). For the aforementioned parameters typical for silicon SWG waveguides, we obtain $n_{xx}=2.79$ and $n_{zz}=1.94$, where $n_{xx}=n_{yy}$ from symmetry considerations. The $k$ vector of a plane wave propagating through such a homogeneous anisotropic uniaxial medium can be calculated using the well-known dispersion equations

We will now show that this behavior can be extended to SWG waveguides, where the periodic structure is finite in the transversal directions [Fig. 3(a)]. Note that when tilting the segments in the SWG waveguide, the period along the propagation direction (${\mathrm{\Lambda }}_z$) is given by ${\mathrm{\Lambda }}_z=\mathrm{\Lambda }/\cos (\theta )$. Extending the equivalence between a periodic structure and a uniaxial crystal, we model the tilted SWG waveguide [Fig. 3(a)] as an anisotropic homogeneous waveguide [Fig. 3(b)]. With the coordinate system set by the light propagation along the waveguide ($z$ direction), the tilt angle $\theta$ of the SWG segments results in a rotation of the diagonal permittivity tensor, yielding the nondiagonal tensor $\widetilde{\mathit{ϵ}}$ [30],

 

Fig. 3. (a) Schematic of a tilted SWG multimode waveguide and (b) the corresponding homogeneous anisotropic model. The ${\mathrm{SiO}}_2$ upper cladding is not shown. (c) Real part of the main component of the fundamental TE $(E_x)$ and TM $(E_y)$ modes propagating along an SWG waveguide with a $\theta =30°$ tilt angle. (d) Real part of the main component of the fundamental TE $(E_x)$ and TM $(E_y)$ modes propagating along the homogeneous anisotropic waveguide. (e) Effective index of the fundamental TE and TM modes of the tilted SWG waveguide and the anisotropic homogeneous waveguide at ${\lambda }_0=1.55\mathrm{\mu m}$.

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From Eq. (1) we observe that the laminar periodic structure is dispersive even if the constituent materials are not. If this structural dispersion is taken into account, the model also predicts the wavelength dependence of the periodic structure, as shown in Fig. 4(a) for an example of $\theta =30°$. Examining this wavelength behavior reveals the intrinsic limitation of any homogenization model: although the anisotropic model is very accurate for long wavelengths (small $\mathrm{\Lambda }/\lambda$ ratios), it becomes less accurate when approaching the Bragg regime $({\lambda }_{\mathrm{Bragg}}\sim 1.3\mathrm{\mu m})$. Finally, Fig. 4(b) shows the effective indexes of the guided TE and TM modes for different tilt angles, revealing that the anisotropic model also provides a good estimation for the higher order modes.

 

Fig. 4. (a) Effective index dispersion of the fundamental TE and TM mode of the $\theta =30°$ tilted SWG waveguide. (b) Effective indices of guided modes in the multimode SWG waveguide, for both polarization and different tilt angles $({\lambda }_0=1.55\mathrm{\mu m})$.

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To experimentally validate the proposed structures, a set of tilted SWG waveguides was fabricated in a 220 nm SOI platform with 3 μm buried oxide (BOX). A 3 μm thick upper ${\mathrm{SiO}}_2$ cladding was deposited by plasma-enhanced chemical vapor deposition (PECVD). The test structure was a Mach–Zehnder interferometer (MZI) where a tilted SWG waveguide was placed in one arm and a reference straight SWG waveguide was placed in the other arm [Fig. 5(a)]. This architecture allows us to easily measure the group index difference between these two types of SWG waveguides. The tapers between the homogeneous and SWG waveguides in the MZI arms were 30 μm long to avoid transition losses, and the SWG waveguides were 600 μm long to accurately characterize the group index variation; a period of $\mathrm{\Lambda }=0.25\mathrm{\mu m}$ was used. To reduce the influence of fabrication jitter, narrow single-mode SWG waveguides are preferred [31], so a width of $w_c=1\mathrm{\mu m}$ was chosen. Highly efficient input and output SWG edge couplers [7,13] were used to couple light in and out of the chip with a lensed polarization maintaining fiber (PMF).The PMF is mounted on a rotatory mount, allowing us to control the input polarization (TE—horizontal or TM—vertical) by using a Glan–Thompson polarizer. The transmission spectrum of each MZI [Fig. 5(b), inset] was measured by sweeping the wavelength of a tunable laser while measuring output power. From the measured transmittance function the group index difference $(\mathrm{\Delta }{n}_g)$ is calculated using the procedure described in Ref. [32]. A comparison to the FDTD simulation results [Fig. 5(b)] shows an excellent agreement for both polarization. By simulating the relation between the group index and effective index in both waveguides, the effective index difference can be estimated from the measured group index difference. In particular, for the fabricated single-mode 45° tilt SWG waveguide we find an effective index variation of $\mathrm{\Delta }{n}_{\mathrm{eff}}^{\mathrm{TE}}\sim 0.23$ for the TE mode and only $\mathrm{\Delta }{n}_{\mathrm{eff}}^{\mathrm{TM}}\sim 0.01$ for the TM mode.

 

Fig. 5. (a) Schematic of the MZI used for the characterization of the tilted SWG waveguides. Insets: SEM images of the SWG tapers and waveguides. (b) Measured and simulated group index difference between the waveguides in the MZI arms. Inset: MZI transmittance measurement for a $\theta =37°$ tilt angle.

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In conclusion, we have shown through both simulation and experiments that tilted SWG structures enable engineering of the effective metamaterial anisotropy in nanophotonics waveguides. Tilting the SWG segments strongly affects both fundamental and higher order TE modes, while this effect is significantly less prominent for TM modes. This approach provides a new degree of freedom to engineer the refractive index in silicon waveguides, while circumventing small variable duty cycles that hamper fabrication. Furthermore, a simple yet accurate homogenization model, based on Eqs. (1)–(5), has been developed and used to characterize and design these structures. The control over metamaterial anisotropy provided by this approach offers promising prospects for advanced polarization management devices and the implementation of metamaterials for transformation optics.